Show-through correction for two-sided documents

ABSTRACT

An image processing method for mitigating the effects of show-through in scanning a duplexed document in a duplex scanning system, including: deriving a scanned first side image P from a first side image A of a duplexed document and a scanned second side image Q from a second side image B of the duplexed document, at least the first side image P including a component thereof attributable to show-through from the second side of the document, and storing images P and Q; from second side image Q, generating a representation thereof corresponding to the show-through contribution to first side image P; processing image P to remove show-through as a function of the representation Q, thereby deriving correct image A.

FIELD OF THE INVENTION

The present invention is directed to the reduction or removal of imagedefects attributable to show-through in two sided image scanningprocess.

BACKGROUND OF THE INVENTION

Acquisition of information adaptable for electronic processingapplications directly from printed original documents has become greatlydesirable as an adjunct to electronic document production. An importantreason for its desirability is the subsequent capability of manipulationof the electronically stored information for editing, compiling andusing the information in forms other that that in which it wasoriginally available. Accordingly, it is desirable to have an imageinformation input capability for such information. In addition to theproduction of new documents from electronically stored information, itwill also be appreciated that some copying functions available in lightlens-type copiers, copying images directly from original documents, maybe more readily accomplished if image information is availableelectronically. Thus, with the capability of electronic input ofinformation, coupled with available output devices, functions such asduplex copying, image rotation, cropping, editing, etc, are possiblewithout the requirement of difficult mechanical manipulation oforiginals and copies.

In duplex scanning, the scan of one side of a typical two-sided piece oftranslucent paper will contain information from both sides of the page.The high contrast information of the front side will be combined with alow contrast version of the image from the back side of the page. Thislow contrast information from the back side of the page is called"show-through".

One method for mechanically reducing show-through is to place a blackpage on the back side of the page while scanning. The light that goesthrough the page is absorbed by the black backing and, although there isa significant reduction of the show-through by this method, there is asmall residual low contrast image of the back side remaining in thescanned image due to light scattering off the back side of the paper.

There are circumstances for which it is not possible to mechanicallyremove show-through. One is when both sides of the page are scannedsimultaneously. In this case, both sides of the page are illuminated andimaged at the same time. Under these conditions, the information on eachside will show through to the opposite side. Another circumstance iswhen a post-scanning correction for show-through is needed for an imagescanned on a scanner with a white cover.

The problem of show-through has sometimes been addressed mechanically,as in U.S. Pat. No. 4,743,974 to Lockwood, which shows simultaneousduplex scanning of duplexed documents, with sensors that are slightlyoffset. The close placement of sensors for simultaneous duplex scanningis desirable for space consideration reasons.

With only the scanned image from one side, there is no way ofdistinguishing between the low contrast "show-through" information fromthe back side and the low contrast information from the front side ofthe page. Additional information is needed to be able to distinguishbetween the two types of low contrast information.

References described hereinabove and below are hereby incorporated byreference for their teachings

SUMMARY OF THE INVENTION

In accordance with the invention, an image processing method isdescribed in which a document is processed for the removal of secondside show-through, for the improvement of image appearance.

In accordance with one aspect of the invention, there is provided animage processing method for mitigating the effects of show-through inscanning a duplexed document in a duplex scanning system, including:deriving a scanned first side image P from a first side image A of aduplexed document and a scanned second side image Q from a second sideimage B of the duplexed document, at least the first side image Pincluding a component thereof attributable to show-through from thesecond side of the document, and storing images P and Q; from secondside image Q, generating a representation thereof corresponding to theshow-through contribution to first side image P; processing image P toremove show-through as a function of the representation Q, therebyderiving correct image A.

In accordance with another aspect of the invention, a representation ofQ is generated by reverse reading image Q and translating image Q interms of spatial coordinates required to bring each pixel of image Q toa position corresponding to the contribution thereof to image P.

With only the scanned image from one side, there is no way ofdistinguishing between the low contrast "show-through" information fromthe back side and the low contrast information from the front side ofthe page. Additional information is needed to be able to distinguishbetween the two types of low contrast information. By scanning bothsides of the page, it is possible obtain the extra information needed.Thus, show-through can be substantially reduced by removing imagecomponents in the scanned image attributable to show-through, by 1)knowing the image content of both first and second sides of the page, 2)knowing the relative alignment of the first and second side imagecontent, and 3) identifying low contrast image portions in the scannedimage attributable only to second side image content.

In this manner, images can be automatically cleaned of show-through withminimal operator intervention, and minimal image impact.

These and other aspects of the invention will become apparent from thefollowing descriptions to illustrate a preferred embodiment of theinvention read in conjunction with the accompanying drawings in which:

FIG. 1 shows a duplex scanner;

FIGS. 2 and 3 show a duplex scanning assembly;

FIG. 4 shows a document processing system in which the present inventionmight be implemented;

FIGS. 5A-5E show a step-by-step illustration of the image as theinvention operates on it; and

FIGS. 6A and 6B are a functional block diagram of a system implementingthe invention.

DETAILED DESCRIPTION OF THE INVENTION

Referring now to the drawings where the showings are for the purpose ofillustrating a preferred embodiment of the invention, and not forlimiting same, FIG. 1 shows a raster input scanner 10 in accordance withthe invention for generating an electronic representation of an image ona document. As used hereinafter, "document" refers to an image bearingoriginal from which copying is desired. Documents may comprise imagesupporting substrates, such as cut sheets of paper, transparencies orother individual pieces of flimsy material, unburst computer formsforming a continuous length of material, or any image bearing substratefrom which copying is desired. When necessary, the type of document forwhich the description finds use will be described with greaterparticularity. For this invention, we are primarily concerned with"duplex" documents, i.e., original documents with images on both facesthereof. "Image" as used herein refers to the information on thedocument, for which scanning is desired, and which will be in somemanner copied to another substrate or to an electronic storage medium."Scanning" as used hereinafter, refers to the relative movement ofphotosensitive sensors with respect to documents for the purpose ofacquiring an electronic representation of the image or information aboutthe image.

Reference is made to U.S. Pat. No. 4,743,974 to Lockwood, and U.S. Pat.No. 4,967,233 to Buchar et al., which describe an example duplex imagescanner in which the invention would find use. Similar problems would benoted in the arrangements described in U.S. Pat. No. 4,673,285 toShogren, U.S. Pat No. 4,734,742 to Klumpp et al. Show-through can be aproblem in any scanner. However, processing can be simplified if secondside image acquisition can occur substantially close in time to firstside image acquisition, as will become apparent.

Raster input scanner 10 may be comprised of upper and lower framemembers 12 and 14. Upper frame member 12 is provided with a documentinput tray 20 for holding a stack of documents D such as cut sheets ofpaper face-up in the document tray from which data acquisition isdesired. Document input tray 20 is provided with a document feeder 22for feeding documents D out from document input tray 20. In a preferredembodiment of the invention, document feeder 22 may be a well-known topfeeding vacuum corrugated feeder found in standard document feeders forlight-lens type copiers for feeding documents in a 1-N order. Documentinput tray 20 may also be upwardly biased or provided with a trayelevator (not shown) to provide documents D in position for feeding.

Documents D are directed by document feeder 22 to an input sheet path 24to be directed to a scanning position. Along sheet path 24, documents Dare driven along by nip roll 26, comprising a pair of rubber rollerswhich drive the documents in a forward direction along the sheet path24. A second nip roll 28 may advantageously comprise a cross rolldeskewing nip to provide documents in a desired registered positionbefore entering the scanning station. A drive motor (not shown) isprovided for driving the document feeder, and drive rollers in the nips,as well as the other driving elements of raster input scanner 10.

Documents 10 are driven along input sheet path 24 into scanning station30. Best viewed at FIGS. 2 and 3, and in accordance with the invention,scanning station 30 is comprised of upper and lower scanning elements 32and 34. Upper scanning element 32 is fixed in position with respect tosheet path 24, and comprises an upper scanning element frame generallyindicated as 36. As better shown in FIG. 3, frame 36 extendstransversely across and above the path of document travel, for thesupport of the scanning array for scanning a document passingthereunder. Referring again to FIGS. 2 and 3, on an upper portion offrame 36, scanning array 38 is supported on scanning array supportmember 40, transverse to the path of sheet travel thereby for scanningdocuments. Scanning array support member 40 is a relatively rigid memberfixed to the frame 36 by a fastening arrangement, and supportingscanning array 38 in position across the path of document travel,without sagging or twisting which could cause imaging imperfections.Scanning array 38 may be a linear array of photosensitive sensors suchas CCD's or photodiodes which are controlled to sense light reflectedfrom a document during an integration period. The photosensitive sensorsdevelop a charge indicative of the amount of light detected, fortransmission to an image processor for use in assimilating anelectronically stored representation of the image on the document. In apreferred embodiment of the invention, the array may comprise severalsmaller arrays butted at their ends to each other to form an unbrokenarray, as described in U.S. Pat. No. 4,604,161 to Araghi. Other types ofsensors are well within the contemplation of the invention.

Documents directed past scanning element 32 are illuminated by a lampassembly supported on frame 36 and comprising lamp 42, and lamp cover 44having aperture 45 engaged to a power supply (not shown) and directinglight towards the document to be scanned. Light from the illuminateddocument is reflected to lens 46, comprising a bundle of imagetransmitting fiber lenses produced under the tradename of SELFOC byNippon Sheet Glass Company Limited, provided to direct light reflectedfrom the document to scanning array 38. Adjacent to the portion of lens46 closest to the document is a frame extension member 48, extendinggenerally towards the document which may be advantageously provided witha light reflecting concave surface 50 to further aid in directing lightfrom lamp 42 to illuminate the document. It will, of course, beappreciated that other optical and illuminating systems may be used toilluminate and direct light from the document to the scanning array 38.

Referring again to FIG. 2, lower scanning element 34 is provided as amirror image of scanning element 32, including a frame 60, scanningarray support member 62, scanning array 64, lamp assembly including lamp66 and lamp cover 68 having aperture 69, lens 70 and frame extensionmember 72 having a reflective surface 74. The primary difference betweenthe two scanning elements is that the lower scanning element 34 isarranged in a position opposing that of upper scanning element 32, inorder to scan images on a side of the document opposite to that scannedby the upper scanning element 34. Accordingly, as viewed in FIGS. 2 and3, images on the upper face of documents, passing through the scanningstation 30 are illuminated and scanned by upper scanning element 32,while images on the lower face of the document are illuminated andscanned by the lower scanning element 34. In practice, it is preferableto have the elements slightly offset from one another so that theillumination provided for scanning one side of the document does notcause shadows detectable by the sensors in the scanning arrays toshow-through the documents. However, the problem of show-through stillexists.

Documents entering the scanning station are driven into and through thestation by a four roll Controlled Velocity Transport (CVT) arrangement80 comprising two sets of nip roll pairs 82 and 84 located at the entryand exit to the scanning station, respectively. Each nip roll pair iscomprised of an upper drive roll 82a and 84a and a lower idler roll 82band 84b. Drive rolls 82a and 84a are driven by a motor (not shown) whichmaintains the speed of the rolls and accordingly, the speed of documentsmoved by the rolls at a carefully controlled constant velocity. Therollers comprising the CVT may be supported separately from the scanningstations at respective positions on the upper and lower frame portions.

With reference to FIGS. 1 and 2, sheets transported through scanningstation 32 are directed between platen glass members 90 and 92 supportedrespectively on upper and lower frame members 12 and 14 which serve tomaintain flatness in documents advancing therethrough for scanning.

As documents are advanced through scanning station 30, photosensitivesensors on scanning arrays 38 and 64 derive an electronic representationof the image thereon for transmission to an image processing or storagedevice.

With reference again to FIG. 1, upon leaving the scanning station 32,documents are driven by CVT 80 to an exit sheet path 94 wherealongdocuments are driven by nip roll pairs 96, 98 and 100 into an outputtray 102. The passage of documents along the described sheet path,having a generally U-shape, with a single fold, and feeding thedocuments in the input tray 20 from the top of the input stack, providesa single natural inversion of the documents, so that they are arrangedface down in output tray 102 in the same order that they were originallyprovided.

Control of the described arrangement may be accomplished in a variety ofways, depending on the planned usage of the device. Feeding ofdocuments, control of sheets directed through the sheet path operationduring the desired modes of operation, including movement of thescanning element 34 in its scanning motion across the platen may becontrolled by an on-board microprocessor of the type commonly used inthe control of light lens copiers, or prior raster scanners and tailoredto the application required by the present device. Control of theimaging assembly may also be an on-board microprocessor device whichrepeatedly polls the sensors of sensor arrays 38 and 64 to derive chargeinformation indicative of the image on documents scanned thereby,assigns the derived data with address information and provides theaddressed information as an output. The control of the device isresponsive to operator information input. Alternatively, the describeddevice may be partially or completely controlled by another device, suchas for example, a computer, a local or remote workstation or compatibleprinter.

FIG. 4 shows a computer workstation 200, connected via a networkconnection 202 to a text and graphics document storage device 204. Theprocessing steps that will be described herein may take place either onthe workstation, or at a server or processor associated with storagedevice 204. The computer workstation may be associated with a printingsystem 206 for reproduction of documents in hard copy. The computerworkstation may also be associated with the scanning system 208 (aspreviously described). As another alternative, the processing describedmay take place within the scanner itself, if the scanner is equippedwith the appropriate processors. This may well be the case in thesituation of a digital copier, which is equivalent to thecomputer/scanner/printer combination described herein.

With reference now to FIGS. 5A-E, we follow the process step by step.FIG. 5A illustrates the two sided document which the present inventionis concerned with, which may have show-through defects.

FIG. 5B illustrates the results of scanning the document. With only thescanned image from one side, there would be no way of distinguishingbetween the low contrast "show-through" information from the back sideand the low contrast information from the front side of the page.Additional information is needed to be able to distinguish between thetwo types of low contrast information. By scanning both sides of thepage, it is possible obtain the extra information needed.

In accordance with the invention, once both sides of an image have beenacquired, show-through mitigation may occur. While the present inventionis illustrated implemented with a duplex scanner, it will be apparentthat the invention could be implemented with a single sensor array, in asystem which provides a second pass of the document past the sensorarray. Another arrangement might include a high quality sensor array anda low quality sensor array, where information from the low qualitysensor array is used for acquiring the second side image. In yet anotherarrangement, the output of a simplex-only scanner can be processed forshow-through mitigation, using two passes of each document through thescanner to derive the required first and second side image information.

With reference now to FIG. 5C, in performing a show-through mitigation,it is important to notice is that the low contrast show-throughinformation will be reversed, left-to-right, from the high contrastinformation of the other side. Therefore, when comparing the highcontrast and low contrast information from the two sides, it isnecessary to reverse one of the images.

With reference to FIG. 5D, the next requirement is to align the highcontrast and low contrast images. Because the images are acquired atdifferent times, or at different sensors, they are not scanned with thesame spatial reference. Skew alignment, lateral shifts and lineardistortions of the image all may be required. For the simultaneous scancase, these corrections can be determined from the geometry of thescanner, or by a calibration process, which scans a test target, andnotes the spatial displacement of target features. For the case of asoftware correction of an arbitrary scanner output, some human guidanceto the correction algorithm may be required.

At this point in the process, a pixel location on side A can beidentified as potentially having a contribution to its value from sideB, due to knowledge of the location of the image on side B.

Accordingly, low contrast information will be removed from areas of afirst image, corresponding to show-through from the second image.Initially it is noted that the show-through image is relatively lowcontrast in nature, mixed in with a high contrast main image. An exampledocument has two sides, A and B. After 1) reverse reading the image intoa buffer, and 2) aligning each image, the document has been processed sothat each pixel location on side A or B corresponds to the both thefront side or correct image A or B, and the show-through reversed imageB or A, respectively. In this condition, the document is ready for showthrough mitigation.

Models are available which describe the resulting image: the resultingscanned images P and Q might be expressed as:

    P(x)=(A(x)+fB(-x))/(1+f)                                   (1)

    Q=(fA(-x)+B(x))/(1+f)                                      (2)

where f is a constant describing the contrast level of the show-throughportion of the image.

The true images, A and B, are therefore given by:

    A(x)=(P(x)-fQ(-x))/(1-f)                                   (3)

    B(x)=(Q(x)-fP(-x))/(1-f)                                   (4)

Accordingly, each pixel in A or B can be modified appropriately, toderive corrected side A or B with minimal impact on the image A or Brespectively.

This correction uses a linear model. Other embodiments of the mixingmodel are possible. For example, the model could include a blur functionon the information from the back side. The model could also includenon-linear combinations of the two images. In such cases, the correctionformulae, in equations 3 and 4, would be different from those shownhere.

In practice, each pixel in scanned image P or Q is then modified througha logic arrangement implementing equations 3 and 4 to derive a trueimage, A or B, respectively.

FIGS. 6A and 6B illustrate a functional block diagram showing onepossible embodiment of the process described. Referring initially toFIG. 6A, initially, scanned images P (which is the scanned combinationof the side A image and the show-through contribution of side B) andscanned image Q (the scanned combination of the side B image, and theshow-through contribution of side A, which for mitigation purposes isconsidered minimal) are scanned and buffered at memory device 300 and302 respectively. Data describing scanned image Q is initially stored atbuffer 302 in a standard 1-N format. It is however, rewritten to buffer304 in an N-1 format, to reverse read the image. Next, the scanned imageQ data is translated to accomplish spatial correction at imagemanipulation processor 306 by an amount determined in a calibrationprocess, in terms of x, y and θ position, so that each pixel of scannedimage Q approximately correspond to the position of its effect onscanned image P. The reverse read and aligned image is subsequentlyprocessed in accordance with equation 3, to remove from scanned image Pa component of the image attributable to show-through.

FIG. 6B is substantially similar to FIG. 6A, with buffer 320 receivingscanned image Q, buffer 322 receiving scanned image P, buffer 324receiving the reverse read version of scanned image P, and imagemanipulation processor 326 spatially translating scanned image P by anamount required to bring it into alignment with scanned image Q.Equation 4 is applied to scanned images P and Q for the derivation oftrue image B from derived scanned images P and Q.

The described equations can be applied to each pixel either by circuitryor software programmed for accomplishing the mathematical calculation,or more likely, with a look-up table, loaded with likely corrections fora given set of materials and conditions.

The calibration used at image manipulation processors 306, 326 can beeither entered by a user on an image by image basis, or derived aninserted automatically with a calibration process that scans a testtarget and locates a target feature with respect to each scanningposition. Likewise, a value of the show-through coefficient f can beentered on a trial and error basis with a user observing the results orby measuring show-through based on known imaging parameters.

Different types of paper will have different amounts of show-through.The type of marking, i.e. toner, ink, pen, etc., may also influence theamount of show-through. These different values will not be radicallydifferent for normal plain paper. If necessary, a manual adjustment tothe amount of show-through correction, i.e. the value of "f", could beprovided to allow the operator to maximize the correction.

The disclosed method may be readily implemented in software using objectoriented software development environments that provide portable sourcecode that can be used on a variety of computer or workstation hardwareplatforms. Alternatively, the disclosed image processing system may beimplemented partially or fully in hardware using standard logic circuitsor specifically on a single chip using VLSI design. Whether software orhardware is used to implement the system varies depending on the speedand efficiency requirements of the system and also the particularfunction and the particular software or hardware systems and theparticular microprocessor or microcomputer systems being utilized. Theimage processing system, however, can be readily developed by thoseskilled in the applicable arts without undue experimentation from thefunctional description provided herein together with a general knowledgeof the computer arts.

While this invention has been described in conjunction with a preferredembodiment thereof, it is evident that many alternatives, modifications,and variations will be apparent to those skilled in the art.Accordingly, it is intended to embrace all such alternatives,modifications and variations as fall within the spirit and broad scopeof the appended claims.

I claim:
 1. An image processing method for mitigating the effects ofshow-through in scanning a duplexed document in a duplex scanningsystem, including:deriving a scanned first side image P from a firstside image A of a duplexed document and a scanned second side image Qfrom a second side image B of the duplexed document, at least the firstside image P including a component thereof attributable to show-throughfrom the second side of the document, and storing images P and Q; fromsecond side image Q, generating a representation thereof correspondingto the show-through contribution to first side image P; processing imageP to remove show-through as a function of the representation Q, therebyderiving correct image A.
 2. The image processing method of claim 1,wherein the representation of Q is generated by reverse reading image Q.3. The image processing method of claim 1, wherein the representation ofQ is generated by translating image Q in terms of spatial coordinates tobring each pixel of image Q to a position corresponding to thecontribution thereof to image P.
 4. The image processing method of claim1, wherein the representation of Q is generated by:reverse reading imageQ; translating reverse read image Q in terms of spatial coordinatesrequired to bring each pixel of image Q to a position corresponding tothe contribution thereof to image P.
 5. The image processing methoddescribed in claim 1, wherein said image P is additionally a function ofa show through coefficient f, indicating the intensity of the showthrough portion of the image.
 6. The image processing method of claim 5,wherein the processing function is given by

    A(x)=(P(x)-(f×Q(-x)))/(1-f).


7. An image processing arrangement for mitigating the effects ofshow-through in scanning a duplexed document in a duplex scanningsystem, including:a scanner deriving a scanned first side image P from afirst side image A of a duplexed document and a scanned second sideimage Q from a second side image B of the duplexed document, at leastthe first side image P including a component thereof attributable toshow-through from the second side of the document, and storing images Pand Q; an image manipulation circuit generating a representation of Qcorresponding to the show-through contribution to first side image P;means for processing image P to remove show-through as a function of therepresentation Q, thereby deriving correct image A.
 8. The imageprocessing arrangement of claim 7, wherein said image manipulationcircuit for generating the representation of Q includes means forreverse reading image Q.
 9. The image processing method arrangement ofclaim 7, wherein the image manipulation circuit generating arepresentation of Q included means for translating image Q in terms ofspatial coordinates to bring each pixel of image Q to a positioncorresponding to the contribution thereof to image P.
 10. The imageprocessing arrangement of claim 7, wherein the image manipulationcircuit generating a representation of Q includes:means for reversereading image Q; and means for translating reverse read image Q in termsof spatial coordinates required to bring each pixel of image Q to aposition corresponding to the contribution thereof to image P.
 11. Theimage processing arrangement described in claim 7, wherein means forprocessing image P additionally takes into account a function of a showthrough coefficient f, indicating the intensity of the show throughportion of the image.
 12. The image processing arrangement of claim 11,wherein the processing means operates in accordance with the functiongiven by

    A(x)=(P(x)-(f×Q(-x)))/(1-f).


13. An image processing method for mitigation the effects ofshow-through in scanning duplexed documents in a duplex scanningsystems, including:scanning a first side image A of a document toacquire an image P including show-through effects of a second side imageB; scanning the second side image B of a document to acquire an image Qincluding the show-through effects of a first side image A; processingimage P for removal of show-through effects, by:generating arepresentation of Q corresponding to the show-through contribution tofirst side image P, processing image P to remove show-through as afunction of the representation Q, thereby deriving correct image A;processing image Q for removal of show-through effects, by:generating arepresentation of P corresponding to the show-through contribution tosecond side image Q, processing image Q to remove show-through as afunction of the representation P, thereby deriving correct image B. 14.The image processing method of claim 13, wherein the representation of Qis generated by reverse reading image Q.
 15. The image processing methodof claim 13, wherein the representation of P is generated by reversereading image P.
 16. The image processing method of claim 13, whereinthe representation of Q is generated by translating image Q in terms ofspatial coordinates to bring each pixel of image Q to a positioncorresponding to the contribution thereof to image P.
 17. The imageprocessing method of claim 13, wherein the representation of P isgenerated by translating image P in terms of spatial coordinates tobring each pixel of image P to a position corresponding to thecontribution thereof to image Q.
 18. The image processing method ofclaim 13, wherein the representation of Q is generated by:reversereading image Q; translating reverse read image Q in terms of spatialcoordinates required to bring each pixel of image Q to a positioncorresponding to the contribution thereof to image P.
 19. The imageprocessing method of claim 13, wherein the representation of P isgenerated by:reverse reading image P; translating reverse read image Pin terms of spatial coordinates required to bring each pixel of image Pto a position corresponding to the contribution thereof to image Q. 20.The image processing method of claim 13, wherein the processing functionis given by

    A(x)=(P(x)-(f×Q(-x)))/(1-f).


21. The image processing method of claim 13, wherein the processingfunction is given by

    B(x)=(Q(x)-(f×P(-x)))/(1-f).


22. A document printing system,including an arrangement for mitigationfor the effects of show-through in reproduction of scanned duplexeddocuments from a duplex scanning systems, including:a scanner deriving ascanned first side image P from a first side image A of a duplexeddocument and a scanned second side image Q from a second side image B ofthe duplexed document, at least the first side image P including acomponent thereof attributable to show-through from the second side ofthe document, and storing images P and Q; an image manipulation circuitgenerating a representation of Q corresponding to the show-throughcontribution to first side image P; means for processing image P toremove show-through as a function of the representation Q, therebyderiving correct image A; printing P, with show-through effectsmitigated.